Problem 3 - Basic Analysis: Introduction to Real Analysis

Question

Prove that if \(\left\\{x_{n}\right\\}\) is a convergent sequence, \(k \in \mathbb{N}\), then \(\lim _{n \rightarrow \infty} x_{n}^{k}=\left(\lim _{n \rightarrow \infty} x_{n}\right)^{k}\) Hint: Use induction.

Step-by-Step Solution

Answer

Use induction to show \(\lim_{n \to \infty} x_n^k = L^k\) for any \(k \in \mathbb{N}\).

Step 1: Sequence Definition and Base Case

Given a convergent sequence \(\left\{x_{n}\right\}\) with limit \(L\), we need to prove that \(\lim_{n \to \infty} x_n^k = L^k\) for natural numbers \(k\). We start by considering the base case, \(k=1\). For \(k=1\), \(x_n^1 = x_n\), and since \(\left\{x_{n}\right\}\) is convergent, \(\lim_{n \to \infty} x_n = L\). Thus, for \(k=1\), \(\lim_{n \to \infty} x_n^k = L^k\). The base case holds.

Step 2: Inductive Hypothesis

Assume the statement is true for some \(k = m\), i.e., \(\lim_{n \to \infty} x_n^m = L^m\). This is our inductive hypothesis.

Step 3: Inductive Step

We need to prove the statement for \(k = m + 1\). Consider \(x_n^{m+1} = x_n^m \cdot x_n\). By the inductive hypothesis, \(\lim_{n \to \infty} x_n^m = L^m\), and by the definition of sequence convergence, \(\lim_{n \to \infty} x_n = L\). Since both sequences converge, the product sequence \(x_n^m \cdot x_n = x_n^{m+1}\) satisfies \(\lim_{n \to \infty} x_n^{m+1} = L^m \cdot L = L^{m+1}\) by properties of limits.

Step 4: Conclusion

By mathematical induction, since the base case and the inductive step both hold, we conclude that \(\lim_{n \to \infty} x_n^k = L^k\) is true for all natural numbers \(k\).

Key Concepts

Mathematical Induction

Mathematical induction is a powerful tool used to prove statements that are true for an infinite number of cases, particularly for natural numbers. It's like a chain reaction—if you prove the first link in the chain (the base case) and the ability for each link to connect to the next (the inductive step), then you've effectively proven the entire chain.

• **Base Case:** You start by showing that a statement holds for the first integer, often n=0 or n=1. In our context, we start by proving that if a sequence \( \{x_n\} \) converges, then \( \lim_{n \to \infty} x_n^1 = L^1 \), which holds since it's just the original sequence.
• **Inductive Step:** Here, you assume the statement is true for some natural number \( n = m \) (this assumption is called the "inductive hypothesis") and then demonstrate it holds for \( n = m + 1 \). In the sequence problem, we assume \( \lim_{n \to \infty} x_n^m = L^m \) and need to show \( \lim_{n \to \infty} x_n^{m+1} = L^{m+1} \).

This process allows you to bridge from specific cases you know to an infinite number of cases that you don't need to check individually. Once the base case and the inductive step are proven, the statement is validated for all natural numbers.

Limit of a Sequence

The limit of a sequence is an essential concept in calculus and analysis. It describes the behavior of a sequence as the index tends toward infinity. Essentially, a sequence \( \{x_n\} \) converges to a limit \( L \) if, as \( n \) increases, \( x_n \) gets arbitrarily close to \( L \).

• A sequence \( \{x_n\} \) is said to converge to the limit \( L \) if for every positive number \( \epsilon \), there exists a positive integer \( N \) such that for all \( n \geq N \), the terms of the sequence satisfy \( |x_n - L| < \epsilon \).
• In our problem, we begin with a convergent sequence \( \{x_n\} \) where \( x_n \) tends to \( L \). The objective is to demonstrate that raising the sequence to a power \( k \) preserves this convergence property.

Understanding limits is critical because they form the basis for continuity, derivatives, and integrals in calculus. Convergent sequences illustrate how sequences behave at the "infinity end," serving as the foundation for more complex analytical concepts.

Properties of Limits

The properties of limits are rules that simplify the process of finding limits. They apply to sequences, functions, and real numbers, and assist in proving statements about limits elegantly.
Some important properties of limits include:

• **Sum Rule:** If \( \lim_{n \to \infty} a_n = A \) and \( \lim_{n \to \infty} b_n = B \), then \( \lim_{n \to \infty} (a_n + b_n) = A + B \).
• **Product Rule:** Using in our step-by-step solution, if both sequences converge, the product also converges:\( \lim_{n \to \infty} (a_n \cdot b_n) = A \cdot B \).
• **Power Rule:** If \( \lim_{n \to \infty} x_n = L \) and \( k \) is a natural number, then \( \lim_{n \to \infty} x_n^k = L^k \).

These properties allow us to extend the results of convergent sequences to more complex expressions. By applying the product rule and understanding base and power relationships, we can see why \( \lim_{n \to \infty} x_n^{m+1} = L^{m+1} \) in our proof, thus concluding that powers of convergent sequences retain their limit convergence.